In this paper lattice models
for strain-gradient elasticity of continuum are suggested.
The first advantage of the suggested lattice models
is a possibility to consider these models
as a microstructural basis of unified description
of gradient models with positive and negative signs
of the strain gradient terms.
A second advantage of the proposed model is that
it can be easily generalized to the case of
the high-order gradient elasticity by using the
other even values of \(s\) .
Using (REF ) with positive integer \(s=2j\) ,
we have non-local interaction of the lattice particle,
that gives the derivatives of integer order \(2 j\) 
in the continuum equation.
Three-dimensional lattice models
and the correspondent continuum equation can also be formulated
as (REF ) and (REF ).
The third advantage of the proposed form of the interaction
is that the lattice equations can be used not only
for the integer but also for fractional values
of the parameter \(s\) .
Therefore the suggested general lattice model can be
extended on the fractional nonlocal case.
The suggested types (REF ) and (REF ) of inter-particle
interactions in the lattice can be used for non-integer \(s\) .
If we consider interaction terms defined by (REF )
and (REF ) with non-integer \(s=\alpha \) ,
then we will get continuum equations with the Riesz
fractional derivatives [1]} of orders \(s=\alpha \) 
by the methods suggested in [2]}, [3]}.
The lattice models with long-range interactions
of the types (REF ) and (REF )
with non-integer \(s=\alpha \) ,
can serve as microscopic models for elastic continuum
with power-law non-locality.
It allows us to derive fractional generalizations
of gradient elasticity by using
a microscopic approach [4]}, [5]}.
We also assume that the suggested lattice model can be
generalized to get discrete (lattice) models for
dislocations in the gradient elasticity continuum
[6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]},
and then it will be possible to extend them
to the fractional non-local case.
The suggested lattice models with long-range interactions,
which are suggested for the gradient elasticity continuum,
can be important to describe the non-local elasticity
of materials
at micro and nano scales [14]}, [15]}, [16]}, [17]},
where the interatomic and intermolecular interactions
are prevalent in determining the properties of these materials.
